In this question we determine that the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges, where the sum runs over primes.
As I see the convergence is really slow. The partial sums for given $N$ finite upper limits are
$\begin{align} N \quad & \text{partial sum}\\ 100 \quad & 0.757042464018193\\ 1000 \quad & 0.803993788114564\\ 10000 \quad & 0.828779261095689\\ 100000 \quad & 0.844238045700797\\ 1000000 \quad & 0.854866046633956\\ \end{align}$
Upto the $1000000$th partial sum there is no significant digit. Could anyone give me the sum of this series for some significant digits? As many as you can, but at least $10$ digits would be nice.
Edit. My calculations above have an $1/(2 \ln 2)$ difference, because the sum runs from $p_2$.
Your sums seem wrong; they are all off by an additive factor of 1/(2 log 2). Why you omitted the prime 2 is unknown to me. Also by N you seem to mean "compute the sum up to and including the N'th prime".
In any event, you can find the value of this sum to about 45 digits here:
https://oeis.org/A137245